It seems you were looking for a name to describe any number in the range [0,1], not the range itself. To say, 'a number in the unit interval' is a bit of a mouthful. Surely there's a better English term for such a common programming requirement? I'll have a go…
Since we're talking about floating point numbers, which the programmer expresses with a finite (and therefor rational) decimal representation, these numbers are a subset of 'decimal fractions'. If the absolute value of the fraction is less than 1 (i.e. the numerator is less than the denominator), this is called a 'proper fraction'. And since we're not interested in negative numbers, we'd have to further clarify these as 'positive' numbers.
So, putting all that together, we could call these numbers positive proper decimal fractions.
Hmm… By word-count, it's a small improvement on 'a number in the unit interval', but it conveniently overlooks the cases of 0 and 1, which are not really 'proper' fractions. Sigh.
Okay, so I'm going to propose the term unidecimal. There, done. Yes, I made it up myself, but hey… these words are all made up by someone at some point!